This article was first written in May 2005 for the BeezNest technical
website (http://glasnost.beeznest.org/articles/256). As of 2008, the most famous JavaScript libraries
are probably Dojo, JQuery and Prototype, but others like MeeBoo, Sarissa and Scriptaculous 
should be considered as well.
Dithered.com - Javascript Library
This is a collection of scripts that I've created and/or modified. All  the scripts are released under a Creative Commons License. Do whatever  you want with them - use them as is or modify them to suit your purposes.  Both personal and commercial usage is ok by me.
Unless otherwise  noted, all scripts degrade gracefully in older browsers that don't support  the functionality the script provides (ie. no error messages are triggered)  and, with the obvious exception of the Browser Detect script, all use object  detection instead of browser detection (which ensures a broad range of  browsers will run these scripts including future browsers).
Website: http://www.dithered.com/javascript/index.html
Scripts: 1k DHTML API, Acrobat (PDF) Detection / Redirection,  Browser Detect, Cookie API, Countdown Timer, Form Functions, Form Validation,  Image Preloader, Quicktime Detection / Redirection, Rollovers, Window Opener  Functions, Window Properties.
DOM Support / Extensions: Array Extensions, DOM1 Core, DOM2 Events, DOM2 Traversal, DOM2 Views, DOM  Extensions.
Scott Andrew - The JavaScript Junkyard
The JavaScript Junkyard is a bunch of silly code I've found useful for  one reason or the other. Some of them are truly useful, such as the cookie  functions, while others are more like adventuresome like inPoly. Many of  the geometry and trig-based functions were designed as components for arcade-style  DHTML games.
Website: http://www.scottandrew.com/weblog/jsjunk Scripts: sortByProperty, addEvent / RemoveEvent, Cookie Functions, getQueryArgs,  getRandomColor, readFile, roundTo, getDistance, getAngle, inPoly.
DomAPI
DomAPI is an application-centric development environment targeted at version  5.0 or better browsers running on Windows, MacOS and Linux. Supported browsers  include Internet Explorer, Netscape Navigator, Mozilla, Safari, Chimera  and Opera.
Website: http://www.domapi.com/
Licence: Versions <= 3: LGPL; >= 4: ?.